\(\int \frac {\text {arccosh}(a x)^3}{(c-a^2 c x^2)^{5/2}} \, dx\) [251]

Optimal result

Integrand size = 22, antiderivative size = 413 \[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=-\frac {x \text {arccosh}(a x)}{c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{2 a c^2 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}+\frac {x \text {arccosh}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 x \text {arccosh}(a x)^3}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^3}{3 a c^2 \sqrt {c-a^2 c x^2}}-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2 \log \left (1-e^{2 \text {arccosh}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \log \left (1-a^2 x^2\right )}{2 a c^2 \sqrt {c-a^2 c x^2}}-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \operatorname {PolyLog}\left (3,e^{2 \text {arccosh}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}} \]

[Out]

1/3*x*arccosh(a*x)^3/c/(-a^2*c*x^2+c)^(3/2)-x*arccosh(a*x)/c^2/(-a^2*c*x^2+c)^(1/2)+2/3*x*arccosh(a*x)^3/c^2/(
-a^2*c*x^2+c)^(1/2)+1/2*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/c^2/(-a^2*x^2+1)/(-a^2*c*x^2+c)^(1/2)+2/3
*arccosh(a*x)^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/c^2/(-a^2*c*x^2+c)^(1/2)-2*arccosh(a*x)^2*ln(1-(a*x+(a*x-1)^(1/2
)*(a*x+1)^(1/2))^2)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/c^2/(-a^2*c*x^2+c)^(1/2)+1/2*ln(-a^2*x^2+1)*(a*x-1)^(1/2)*(a
*x+1)^(1/2)/a/c^2/(-a^2*c*x^2+c)^(1/2)-2*arccosh(a*x)*polylog(2,(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)*(a*x-1)^(
1/2)*(a*x+1)^(1/2)/a/c^2/(-a^2*c*x^2+c)^(1/2)+polylog(3,(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)*(a*x-1)^(1/2)*(a*
x+1)^(1/2)/a/c^2/(-a^2*c*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {5901, 5899, 5913, 3797, 2221, 2611, 2320, 6724, 5912, 5914, 5900, 266} \[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=-\frac {2 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {a x-1} \sqrt {a x+1} \operatorname {PolyLog}\left (3,e^{2 \text {arccosh}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}+\frac {2 x \text {arccosh}(a x)^3}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^3}{3 a c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{2 a c^2 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}-\frac {x \text {arccosh}(a x)}{c^2 \sqrt {c-a^2 c x^2}}-\frac {2 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2 \log \left (1-e^{2 \text {arccosh}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}+\frac {x \text {arccosh}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {\sqrt {a x-1} \sqrt {a x+1} \log \left (1-a^2 x^2\right )}{2 a c^2 \sqrt {c-a^2 c x^2}} \]

[In]

Int[ArcCosh[a*x]^3/(c - a^2*c*x^2)^(5/2),x]

[Out]

-((x*ArcCosh[a*x])/(c^2*Sqrt[c - a^2*c*x^2])) + (Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/(2*a*c^2*(1 - a^
2*x^2)*Sqrt[c - a^2*c*x^2]) + (x*ArcCosh[a*x]^3)/(3*c*(c - a^2*c*x^2)^(3/2)) + (2*x*ArcCosh[a*x]^3)/(3*c^2*Sqr
t[c - a^2*c*x^2]) + (2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3)/(3*a*c^2*Sqrt[c - a^2*c*x^2]) - (2*Sqrt[-1
+ a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2*Log[1 - E^(2*ArcCosh[a*x])])/(a*c^2*Sqrt[c - a^2*c*x^2]) + (Sqrt[-1 + a*x
]*Sqrt[1 + a*x]*Log[1 - a^2*x^2])/(2*a*c^2*Sqrt[c - a^2*c*x^2]) - (2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]
*PolyLog[2, E^(2*ArcCosh[a*x])])/(a*c^2*Sqrt[c - a^2*c*x^2]) + (Sqrt[-1 + a*x]*Sqrt[1 + a*x]*PolyLog[3, E^(2*A
rcCosh[a*x])])/(a*c^2*Sqrt[c - a^2*c*x^2])

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 3797

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> Simp[(-I)*((
c + d*x)^(m + 1)/(d*(m + 1))), x] + Dist[2*I, Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*
fz*x))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 5899

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[x*((a + b*ArcCosh
[c*x])^n/(d*Sqrt[d + e*x^2])), x] + Dist[b*c*(n/d)*Simp[Sqrt[1 + c*x]*(Sqrt[-1 + c*x]/Sqrt[d + e*x^2])], Int[x
*((a + b*ArcCosh[c*x])^(n - 1)/(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ
[n, 0]

Rule 5900

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(((d1_) + (e1_.)*(x_))^(3/2)*((d2_) + (e2_.)*(x_))^(3/2)), x_Sym
bol] :> Simp[x*((a + b*ArcCosh[c*x])^n/(d1*d2*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])), x] + Dist[b*c*(n/(d1*d2))*Sim
p[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]], Int[x*((a + b*ArcCosh[c*x])^(n - 1)/(1
- c^2*x^2)), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0]

Rule 5901

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*(d + e*x^2)^(
p + 1)*((a + b*ArcCosh[c*x])^n/(2*d*(p + 1))), x] + (Dist[(2*p + 3)/(2*d*(p + 1)), Int[(d + e*x^2)^(p + 1)*(a
+ b*ArcCosh[c*x])^n, x], x] - Dist[b*c*(n/(2*(p + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)], Int[x*(
1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] &&
EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]

Rule 5912

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(
x_))^(p_.), x_Symbol] :> Int[(f*x)^m*(d1*d2 + e1*e2*x^2)^p*(a + b*ArcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e
1, d2, e2, f, m, n}, x] && EqQ[d2*e1 + d1*e2, 0] && IntegerQ[p]

Rule 5913

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/e, Subst[Int[(
a + b*x)^n*Coth[x], x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]

Rule 5914

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x^2)^
(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e*(p + 1))), x] - Dist[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*
(-1 + c*x)^p)], Int[(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a,
b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps \begin{align*} \text {integral}& = \frac {x \text {arccosh}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{3 c}+\frac {\left (a \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x \text {arccosh}(a x)^2}{(-1+a x)^2 (1+a x)^2} \, dx}{c^2 \sqrt {c-a^2 c x^2}} \\ & = \frac {x \text {arccosh}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 x \text {arccosh}(a x)^3}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {\left (a \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x \text {arccosh}(a x)^2}{\left (-1+a^2 x^2\right )^2} \, dx}{c^2 \sqrt {c-a^2 c x^2}}+\frac {\left (2 a \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x \text {arccosh}(a x)^2}{1-a^2 x^2} \, dx}{c^2 \sqrt {c-a^2 c x^2}} \\ & = \frac {\sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{2 a c^2 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}+\frac {x \text {arccosh}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 x \text {arccosh}(a x)^3}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {\text {arccosh}(a x)}{(-1+a x)^{3/2} (1+a x)^{3/2}} \, dx}{c^2 \sqrt {c-a^2 c x^2}}-\frac {\left (2 \sqrt {-1+a x} \sqrt {1+a x}\right ) \text {Subst}\left (\int x^2 \coth (x) \, dx,x,\text {arccosh}(a x)\right )}{a c^2 \sqrt {c-a^2 c x^2}} \\ & = -\frac {x \text {arccosh}(a x)}{c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{2 a c^2 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}+\frac {x \text {arccosh}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 x \text {arccosh}(a x)^3}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^3}{3 a c^2 \sqrt {c-a^2 c x^2}}+\frac {\left (4 \sqrt {-1+a x} \sqrt {1+a x}\right ) \text {Subst}\left (\int \frac {e^{2 x} x^2}{1-e^{2 x}} \, dx,x,\text {arccosh}(a x)\right )}{a c^2 \sqrt {c-a^2 c x^2}}-\frac {\left (a \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x}{1-a^2 x^2} \, dx}{c^2 \sqrt {c-a^2 c x^2}} \\ & = -\frac {x \text {arccosh}(a x)}{c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{2 a c^2 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}+\frac {x \text {arccosh}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 x \text {arccosh}(a x)^3}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^3}{3 a c^2 \sqrt {c-a^2 c x^2}}-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2 \log \left (1-e^{2 \text {arccosh}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \log \left (1-a^2 x^2\right )}{2 a c^2 \sqrt {c-a^2 c x^2}}+\frac {\left (4 \sqrt {-1+a x} \sqrt {1+a x}\right ) \text {Subst}\left (\int x \log \left (1-e^{2 x}\right ) \, dx,x,\text {arccosh}(a x)\right )}{a c^2 \sqrt {c-a^2 c x^2}} \\ & = -\frac {x \text {arccosh}(a x)}{c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{2 a c^2 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}+\frac {x \text {arccosh}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 x \text {arccosh}(a x)^3}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^3}{3 a c^2 \sqrt {c-a^2 c x^2}}-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2 \log \left (1-e^{2 \text {arccosh}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \log \left (1-a^2 x^2\right )}{2 a c^2 \sqrt {c-a^2 c x^2}}-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}+\frac {\left (2 \sqrt {-1+a x} \sqrt {1+a x}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^{2 x}\right ) \, dx,x,\text {arccosh}(a x)\right )}{a c^2 \sqrt {c-a^2 c x^2}} \\ & = -\frac {x \text {arccosh}(a x)}{c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{2 a c^2 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}+\frac {x \text {arccosh}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 x \text {arccosh}(a x)^3}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^3}{3 a c^2 \sqrt {c-a^2 c x^2}}-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2 \log \left (1-e^{2 \text {arccosh}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \log \left (1-a^2 x^2\right )}{2 a c^2 \sqrt {c-a^2 c x^2}}-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}+\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{2 \text {arccosh}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}} \\ & = -\frac {x \text {arccosh}(a x)}{c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{2 a c^2 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}+\frac {x \text {arccosh}(a x)^3}{3 c \left (c-a^2 c x^2\right )^{3/2}}+\frac {2 x \text {arccosh}(a x)^3}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^3}{3 a c^2 \sqrt {c-a^2 c x^2}}-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2 \log \left (1-e^{2 \text {arccosh}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \log \left (1-a^2 x^2\right )}{2 a c^2 \sqrt {c-a^2 c x^2}}-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \operatorname {PolyLog}\left (3,e^{2 \text {arccosh}(a x)}\right )}{a c^2 \sqrt {c-a^2 c x^2}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 0.90 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.65 \[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\frac {\sqrt {\frac {-1+a x}{1+a x}} (1+a x) \left (-i \pi ^3-\frac {12 a x \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x)}{-1+a x}+\frac {6 \text {arccosh}(a x)^2}{1-a^2 x^2}+8 \text {arccosh}(a x)^3+\frac {8 a x \sqrt {\frac {-1+a x}{1+a x}} \text {arccosh}(a x)^3}{-1+a x}-\frac {4 a x \left (\frac {-1+a x}{1+a x}\right )^{3/2} \text {arccosh}(a x)^3}{(-1+a x)^3}-24 \text {arccosh}(a x)^2 \log \left (1-e^{2 \text {arccosh}(a x)}\right )+12 \log (a x)+12 \log \left (\frac {\sqrt {\frac {-1+a x}{1+a x}} (1+a x)}{a x}\right )-24 \text {arccosh}(a x) \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(a x)}\right )+12 \operatorname {PolyLog}\left (3,e^{2 \text {arccosh}(a x)}\right )\right )}{12 a c^2 \sqrt {c-a^2 c x^2}} \]

[In]

Integrate[ArcCosh[a*x]^3/(c - a^2*c*x^2)^(5/2),x]

[Out]

(Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*((-I)*Pi^3 - (12*a*x*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x])/(-1 + a*x)
+ (6*ArcCosh[a*x]^2)/(1 - a^2*x^2) + 8*ArcCosh[a*x]^3 + (8*a*x*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]^3)/(-1
+ a*x) - (4*a*x*((-1 + a*x)/(1 + a*x))^(3/2)*ArcCosh[a*x]^3)/(-1 + a*x)^3 - 24*ArcCosh[a*x]^2*Log[1 - E^(2*Ar
cCosh[a*x])] + 12*Log[a*x] + 12*Log[(Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x))/(a*x)] - 24*ArcCosh[a*x]*PolyLog[2,
E^(2*ArcCosh[a*x])] + 12*PolyLog[3, E^(2*ArcCosh[a*x])]))/(12*a*c^2*Sqrt[c - a^2*c*x^2])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(954\) vs. \(2(400)=800\).

Time = 1.32 (sec) , antiderivative size = 955, normalized size of antiderivative = 2.31

method	result	size
default	\(-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (2 a^{3} x^{3}-3 a x -2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}+2 \sqrt {a x -1}\, \sqrt {a x +1}\right ) \operatorname {arccosh}\left (a x \right ) \left (6 a^{3} x^{3} \operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}+6 a^{4} x^{4} \operatorname {arccosh}\left (a x \right )+6 a^{3} x^{3} \sqrt {a x -1}\, \sqrt {a x +1}+6 a^{4} x^{4}+6 a^{2} x^{2} \operatorname {arccosh}\left (a x \right )^{2}-9 a x \,\operatorname {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}-12 a^{2} x^{2} \operatorname {arccosh}\left (a x \right )-6 \sqrt {a x -1}\, \sqrt {a x +1}\, a x -18 a^{2} x^{2}-8 \operatorname {arccosh}\left (a x \right )^{2}+6 \,\operatorname {arccosh}\left (a x \right )+12\right )}{6 \left (3 a^{6} x^{6}-10 a^{4} x^{4}+11 a^{2} x^{2}-4\right ) c^{3} a}-\frac {\sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \ln \left (\sqrt {a x -1}\, \sqrt {a x +1}+a x -1\right )}{c^{3} a \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{3} a \left (a^{2} x^{2}-1\right )}+\frac {2 \sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \ln \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{3} a \left (a^{2} x^{2}-1\right )}-\frac {4 \sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (a x \right )^{3}}{3 c^{3} a \left (a^{2} x^{2}-1\right )}+\frac {2 \sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (a x \right )^{2} \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{3} a \left (a^{2} x^{2}-1\right )}+\frac {4 \sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (a x \right ) \operatorname {polylog}\left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{3} a \left (a^{2} x^{2}-1\right )}-\frac {4 \sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \operatorname {polylog}\left (3, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{3} a \left (a^{2} x^{2}-1\right )}+\frac {2 \sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (a x \right )^{2} \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{3} a \left (a^{2} x^{2}-1\right )}+\frac {4 \sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (a x \right ) \operatorname {polylog}\left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{3} a \left (a^{2} x^{2}-1\right )}-\frac {4 \sqrt {a x +1}\, \sqrt {a x -1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \operatorname {polylog}\left (3, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{c^{3} a \left (a^{2} x^{2}-1\right )}\)	\(955\)

[In]

int(arccosh(a*x)^3/(-a^2*c*x^2+c)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/6*(-c*(a^2*x^2-1))^(1/2)*(2*a^3*x^3-3*a*x-2*a^2*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)+2*(a*x-1)^(1/2)*(a*x+1)^(1/
2))*arccosh(a*x)*(6*a^3*x^3*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)+6*a^4*x^4*arccosh(a*x)+6*a^3*x^3*(a*x-1)^
(1/2)*(a*x+1)^(1/2)+6*a^4*x^4+6*a^2*x^2*arccosh(a*x)^2-9*a*x*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)-12*a^2*x
^2*arccosh(a*x)-6*(a*x-1)^(1/2)*(a*x+1)^(1/2)*a*x-18*a^2*x^2-8*arccosh(a*x)^2+6*arccosh(a*x)+12)/(3*a^6*x^6-10
*a^4*x^4+11*a^2*x^2-4)/c^3/a-(a*x+1)^(1/2)*(a*x-1)^(1/2)*(-c*(a^2*x^2-1))^(1/2)/c^3/a/(a^2*x^2-1)*ln((a*x-1)^(
1/2)*(a*x+1)^(1/2)+a*x-1)-(a*x+1)^(1/2)*(a*x-1)^(1/2)*(-c*(a^2*x^2-1))^(1/2)/c^3/a/(a^2*x^2-1)*ln(1+a*x+(a*x-1
)^(1/2)*(a*x+1)^(1/2))+2*(a*x+1)^(1/2)*(a*x-1)^(1/2)*(-c*(a^2*x^2-1))^(1/2)/c^3/a/(a^2*x^2-1)*ln(a*x+(a*x-1)^(
1/2)*(a*x+1)^(1/2))-4/3*(a*x+1)^(1/2)*(a*x-1)^(1/2)*(-c*(a^2*x^2-1))^(1/2)/c^3/a/(a^2*x^2-1)*arccosh(a*x)^3+2*
(a*x+1)^(1/2)*(a*x-1)^(1/2)*(-c*(a^2*x^2-1))^(1/2)/c^3/a/(a^2*x^2-1)*arccosh(a*x)^2*ln(1+a*x+(a*x-1)^(1/2)*(a*
x+1)^(1/2))+4*(a*x+1)^(1/2)*(a*x-1)^(1/2)*(-c*(a^2*x^2-1))^(1/2)/c^3/a/(a^2*x^2-1)*arccosh(a*x)*polylog(2,-a*x
-(a*x-1)^(1/2)*(a*x+1)^(1/2))-4*(a*x+1)^(1/2)*(a*x-1)^(1/2)*(-c*(a^2*x^2-1))^(1/2)/c^3/a/(a^2*x^2-1)*polylog(3
,-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))+2*(a*x+1)^(1/2)*(a*x-1)^(1/2)*(-c*(a^2*x^2-1))^(1/2)/c^3/a/(a^2*x^2-1)*arcc
osh(a*x)^2*ln(1-a*x-(a*x-1)^(1/2)*(a*x+1)^(1/2))+4*(a*x+1)^(1/2)*(a*x-1)^(1/2)*(-c*(a^2*x^2-1))^(1/2)/c^3/a/(a
^2*x^2-1)*arccosh(a*x)*polylog(2,a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))-4*(a*x+1)^(1/2)*(a*x-1)^(1/2)*(-c*(a^2*x^2-1
))^(1/2)/c^3/a/(a^2*x^2-1)*polylog(3,a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))

Fricas [F]

\[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{3}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(arccosh(a*x)^3/(-a^2*c*x^2+c)^(5/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-a^2*c*x^2 + c)*arccosh(a*x)^3/(a^6*c^3*x^6 - 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 - c^3), x)

Sympy [F]

\[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int \frac {\operatorname {acosh}^{3}{\left (a x \right )}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate(acosh(a*x)**3/(-a**2*c*x**2+c)**(5/2),x)

[Out]

Integral(acosh(a*x)**3/(-c*(a*x - 1)*(a*x + 1))**(5/2), x)

Maxima [F]

\[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{3}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(arccosh(a*x)^3/(-a^2*c*x^2+c)^(5/2),x, algorithm="maxima")

[Out]

integrate(arccosh(a*x)^3/(-a^2*c*x^2 + c)^(5/2), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(arccosh(a*x)^3/(-a^2*c*x^2+c)^(5/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arccosh}(a x)^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int \frac {{\mathrm {acosh}\left (a\,x\right )}^3}{{\left (c-a^2\,c\,x^2\right )}^{5/2}} \,d x \]

[In]

int(acosh(a*x)^3/(c - a^2*c*x^2)^(5/2),x)

[Out]

int(acosh(a*x)^3/(c - a^2*c*x^2)^(5/2), x)